Ray's New Practical Arithmetic · Art. 225 · Unit 22: Ratio and Proportion
105. Compound proportion
Goal: Your child solves problems with several conditions (men, days, size of work) by compound proportion.
You'll need
- Paper and pencil, or a slate or small whiteboard
What's in the book
Page 289 begins Compound Proportion (Art. 225) with a worked example. Page 290 works a wall-building example in full and gives the rule. Pages 290–291 have problems with printed answers.
The lesson, step by step
Warm-up
Ask: 2 people earn $20 in 5 days. How much do 6 people earn in 5 days? Now in 10 days?
Work the book's first example
If 2 men earn $20 in 5 da., what sum can 6 men earn in 10 da.?
The answer is dollars, so $20 is the third term. More men earn more: 6 second, 2 first. More days earn more: 10 second, 5 first.
The answer: 6 × 10 × 20 ÷ (2 × 5) = $120.
The wall example
Page 290: 6 men build a wall 20 ft. long, 3 ft. high, 2 ft. thick in 10 days. How long will 15 men take for a wall 80 ft. long, 2 ft. high, 3 ft. thick?
Take each pair alone and ask "more days or fewer?" The book gets 6 × 80 × 2 × 3 × 10 ÷ (15 × 20 × 3 × 2) = 16 days.
The rule
2. Arrange each pair of the numbers forming the compound ratio as if with the third term, they formed a simple proportion.
3. Divide the product of the numbers in the second and third terms by the product of the numbers in the first term; the quotient will be the fourth term or number required.
Cancel before multiplying.
Guided practice
Do these together:
If a man travel 24 mi. in 2 da., by walking 4 hr. a day: at the same rate, how far will he travel in 10 da., walking 8 hr. a day?
If 16 men build 18 rods of fence in 12 days, how many men can build 72 rd. in 8 da.?
Independent practice and check
Your child works problems 5–11 on pages 290–291. The book prints the answer at the end of each problem line. Have your child cover the answers with a strip of paper, work the problem, then slide the strip down to check.